PATTERNS IN NUMBER THEORY Theophanes E. Raptis Computational Applications Group Division of Applied Technologies NCSR Demokritos 2015

Transcription

1 PATTERNS IN NUMBER THEORY Theohaes E. Rats Comutatoal Alcatos Grou Dvso of Aled Techologes NCSR Demokrtos 205

2 WHY NUMBER THEORY? Comuters are based o tegers eve for smulatos of cotuous systems ad yet they ca buld vrtual worlds! Nuclear Sectra have a dstctvely smlar structure wth rme dstrbutos Radom Matrx Theory May Body Chaos). SIC-POVM Q. Crytograhy): Samlgs from the Bloch s ut shere lead to uexected coectos wth t-desg combatorcs.

3 DISCRETE MATHEMATICS Sgfcace of the Axomatc Bass Thk of geometry wthout Eucld!) No Axomatzato Algorthm ossble. What s the smlest set of requred axoms? How ca we kow that we use the best oe? How may alteratves ossble?

4 FRACTALITY IN ARITHMETICS: WHENCE FROM? Varous umber-theoretc fuctos exhbt selfsmlarty. Smlest case: Dgt-Sums all alhabets. Image Source: Wolfram Ste) Smle recurso schemes reroduce the above.

5 PRIMITIVE SELF-SIMILARITY Remsct Sets: Ivarace uder removal of a subset. -ary Tree Ufoldgs: Lexcograhcally Ordered Combatorc Powersets Lexcos ) of symbol alhabets. More rmtve tha Cator Sets. Rows: Perodcty Colums: Quas-erodcty

6 GENERAL QUASI-PERIODIC SYSTEM Let { y y } a set of harmoc fuctos of creasg erods Lalaca egefuctos ). Let h yl ) a threshold samler: { 2πt / τ ) /2} h y t) L) SIGNy L For ay quatzed terval { δτ δτ Ν } we get a set of bary ulse sequeces eg Number theoretc equvalet: mod x+ x )) h x+ x + 0 2) Θ 0 2

7 TRAILING ZEROS AND DIVISIBILITY

8 INTERPOLANTS FOR T Z NDIVISOR) Tz0:2 8 2): [ log2 N )] k kh 2 2k ) [log2 N )] + h 2 2 ) Tz0:3 5 3): [ log3 N )] k kh 3 3k )

9 EXAMPLE: COLLATZ DYNAMICS Brached mag: Folded forms: Equvalet dyamcs: Tz x2) 32 x) + 3x+ Tz3x+ 2) 3x+ )2 [log 2 x)] h..) 3x+ ) 2 2 [log 2 x)] h..) [log 2 x)] h..) 2

10 PRIMALITY VIA TZ Let a lst of rmes <. Equvaletly ) ) 2 )... : 2 k z z z T k T T N { } k... )] [log ) h k kj j IsPrme): No tralg zeros ay base <.)... ) : k k k h k k j N { } k Z T 0 )

11 DIVISOR FUNCTION Def.: Examles: Proertes: σ 0 ) ~ umber of factors of a b 0 t t2 t 2...) ) t σ )... σ σ 2 From the last oe t ) σ 0 ) + 2

12 HIDDEN DIMENSIONALITY From last formula ad Tz: σ 0 ) Z + Exad: T ) ) [log )] k σ ) ) ) k+ h k k h k Trasform to Sum-of-Products: σ [log )] k+ 0 ) k+ ) h k 0 k ) k Products o rhs rereset a Mult-Dm. Bary Grd! k

13 DIMENSIONAL ORDERING OF INTEGERS Samle [0 024]

14 THE DIVISOR FUNCTION AS A DIRECT SUM Smler Exresso: Suerosto of st le of all Lexcos for all alhabet bases Aly logcal mask the above array ~A0) Equvalet: σ 0 ) + h k) k

15 Resultg Matrx 32 x 32) Partcular case of Quas-erodcty. Oe Problem: Do other cases of ostve sequeces of erods exst that exhbt global extrema? Are they assocated wth certa umber theoretc roertes? Check k h k ) j k h j k )

16 FUNDAMENTAL PROPERTY OF D-MATRIX Assocato of Dvsors wth Iteger Parttos. Geometry of the Dvsor Set Let D{ d d k } a lst of dvsors of. ) ) 0 d d d σ {{{ z y x RSA 2-factor rmes smultaeously satsfy: Objectve Fucto: ) d d d N z y x y x x z y x ) ) j j + 3 / 3 /... 3 /

17 Smle Examle 5 z 7): For all other values of z the 2 factors ever meet alog the same colum. The smle structure of 2-restrcted arttos allows a easy ad fast factorzato algorthm.

18 OPTIMIZED VERSION WITH PRECOMPUTED ELEMENTS The Matrx of all 2-Parttos Products s recursve! We ca avod all multlcatos.

19 Numerc structure: Dffereces: k+ ) / 2 + {0} k { 2} < 2

20 NEXUS PRIMUS: A PROJECTIVE GEOMETRY? The Set of dscrete dagoals of the Dvsor Matrx form a Budle uder cotuato. { [ ] ± } y ) ) / /* + )) m m m x

21 Magfed Area ear the ceter shows two eveloes 52)

22 h ): No roots h ): Roots structured.

23 DEVIL STAIRCASES Let yx) a strctly creasg ostve eveloe ad q over y a regular or rregular quatzer. Every such starcase admts a uque vertble decomosto as Q Idetty: { h x x q ) k k)) + h x> x q ) k x q ) } k y) δq ) h x x q ) k k) + h x k x q ) k) Orgal sgal ecoded sequece Cumulat Curve for Dvsor Matrx: δ h x x ) ) q q + ) / 2

24 COMBINATORIC DESIGNS Aalyss of the structure of Powersets shows all desgs to be related wth Partto Fuctos Smle examle: Let Lb6 N0) the sextary Lexco of all 0 symbol strgs. We ask for all the sequeces that cota oly grous of 3 ad 5. Symbol Couter fuctos: h σ N { 0... b } ) h + 0 σ ) b ) b ) σ Characterstc over a Partto Fucto: χ{ σ }{ σ }) M * j e h σ ) σ ) * σ { σ })